10,369 research outputs found
An efficient shooting algorithm for Evans function calculations in large systems
In Evans function computations of the spectra of asymptotically
constant-coefficient linear operators, a basic issue is the efficient and
numerically stable computation of subspaces evolving according to the
associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be
obtained by representing subspaces as single exterior products
\cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the
exterior-product space quickly becomes prohibitive, growing as ,
where is the dimension of the system written as a first-order ODE and
(typically ) is the dimension of the subspace. We resolve this
difficulty by the introduction of a simple polar coordinate algorithm
representing ``pure'' (monomial) products as scalar multiples of orthonormal
bases, for which the angular equation is a numerically optimized version of the
continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial
equation is evaluable by quadrature. Notably, the polar-coordinate method
preserves the important property of analyticity with respect to parameters.Comment: 21 pp., two figure
Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections
We propose new methods for the numerical continuation of point-to-cycle
connecting orbits in 3-dimensional autonomous ODE's using projection boundary
conditions. In our approach, the projection boundary conditions near the cycle
are formulated using an eigenfunction of the associated adjoint variational
equation, avoiding costly and numerically unstable computations of the
monodromy matrix. The equations for the eigenfunction are included in the
defining boundary-value problem, allowing a straightforward implementation in
AUTO, in which only the standard features of the software are employed.
Homotopy methods to find connecting orbits are discussed in general and
illustrated with several examples, including the Lorenz equations. Complete
AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE
system, are freely available.Comment: 18 pages, 10 figure
Query-points visibility constraint minimum link paths in simple polygons
We study the query version of constrained minimum link paths between two
points inside a simple polygon with vertices such that there is at
least one point on the path, visible from a query point. The method is based on
partitioning into a number of faces of equal link distance from a point,
called a link-based shortest path map (SPM). Initially, we solve this problem
for two given points , and a query point . Then, the proposed
solution is extended to a general case for three arbitrary query points ,
and . In the former, we propose an algorithm with preprocessing
time. Extending this approach for the latter case, we develop an algorithm with
preprocessing time. The link distance of a - path between
, as well as the path are provided in time and , respectively, for the above two cases, where is the number of links
The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems
The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem
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